{"title":"The distribution of some sequences connected with the nearest integer continued fraction","authors":"Cor Kraaikamp","doi":"10.1016/S1385-7258(87)80038-0","DOIUrl":null,"url":null,"abstract":"<div><p>Let <em>A</em><sub><em>n</em></sub>/<em>B</em><sub><em>n</em></sub>, n = 1,2,… denote the sequence of convergents of the nearest integer continued fraction expansion of the irrational number <em>x</em>, and defineΘ<sub><em>n</em></sub>(<em>x</em>): <em>B</em><sub><em>n</em></sub>|<em>B</em><sub><em>n</em></sub><em>x</em> − <em>A</em><sub><em>n</em></sub>|, n = 1,2,…. In this paper the distribution of the two-dimensional sequence (Θ<sub><em>n</em></sub>(<em>x</em>), Θ<sub><em>n+1</em></sub>(<em>x</em>)), <em>n</em> = 1,2,… is determined for almost all <em>x</em>.</p><p>Various corollaries are obtained, for instance Sendov's analogue of Vahlen's theorem for the nearest integer continued fraction. The present method is an extension of the work by H. Jager on the corresponding problem for the regular continued fraction expansion.</p></div>","PeriodicalId":100664,"journal":{"name":"Indagationes Mathematicae (Proceedings)","volume":"90 2","pages":"Pages 177-191"},"PeriodicalIF":0.0000,"publicationDate":"1987-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S1385-7258(87)80038-0","citationCount":"14","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Indagationes Mathematicae (Proceedings)","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1385725887800380","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 14
Abstract
Let An/Bn, n = 1,2,… denote the sequence of convergents of the nearest integer continued fraction expansion of the irrational number x, and defineΘn(x): Bn|Bnx − An|, n = 1,2,…. In this paper the distribution of the two-dimensional sequence (Θn(x), Θn+1(x)), n = 1,2,… is determined for almost all x.
Various corollaries are obtained, for instance Sendov's analogue of Vahlen's theorem for the nearest integer continued fraction. The present method is an extension of the work by H. Jager on the corresponding problem for the regular continued fraction expansion.
设An/Bn, n = 1,2，…表示无理数x的最近整数连分式展开的收敛序列，defineΘn(x): Bn|Bnx−An|， n = 1,2，....本文确定了二维数列(Θn(x)， Θn+1(x))， n = 1,2，…对几乎所有x的分布，得到了若干推论，如最近整数连分式的Sendov对Vahlen定理的类比。本文方法是对H. Jager关于正则连分数展开问题的推广。
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